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Page 143
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue measurable ...
... measure on as well as for its extension on * should cause no confusion . The measure of Lebesgue in an interval [ a , b ] of real numbers may be defined as the Lebesgue extension of the Borel measure in [ a , b ] . The Lebesgue measurable ...
Page 188
... measure has the property n = 1 n n μ ( PE1 ) = دد μ ‚ ( E ; ) , i = 1 II i = 1 Ε , Ε Σ . , and it follows from the ... Lebesgue measure on the real line for i = 1 , ... , n . Then S PS , is n - dimensional Eucli- dean space , and μ = μ1 ...
... measure has the property n = 1 n n μ ( PE1 ) = دد μ ‚ ( E ; ) , i = 1 II i = 1 Ε , Ε Σ . , and it follows from the ... Lebesgue measure on the real line for i = 1 , ... , n . Then S PS , is n - dimensional Eucli- dean space , and μ = μ1 ...
Page 223
... Lebesgue measure and that f ' may vanish almost everywhere without ƒ being a constant . 7 Let ( S , E , μ ) be the product of the regular measure spaces ( Sα , Σa , μa ) , α e A. If S is endowed with its product topology show that ( S ...
... Lebesgue measure and that f ' may vanish almost everywhere without ƒ being a constant . 7 Let ( S , E , μ ) be the product of the regular measure spaces ( Sα , Σa , μa ) , α e A. If S is endowed with its product topology show that ( S ...
Contents
Preliminary Concepts | 1 |
B Topological Preliminaries | 10 |
Algebraic Preliminaries | 34 |
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A₁ Acad additive set function algebra Amer analytic arbitrary B-space B₁ ba(S Banach spaces Borel sets ca(S Cauchy sequence compact Hausdorff space compact operator complex numbers contains continuous functions converges convex set Corollary countably additive DEFINITION dense disjoint Doklady Akad E₁ element ergodic exists f₁ finite dimensional function defined function f Hausdorff space Hence Hilbert space homeomorphism implies inequality integral isometric isomorphism K₁ L₁ L₁(S Lebesgue Lemma Let f linear manifold linear map linear operator linear topological space measurable functions measure space metric space Nauk SSSR N. S. neighborhood non-negative o-field o-finite open set operator topology positive measure space Proc PROOF properties proved real numbers reflexive Riesz S₁ scalar semi-group sequentially compact Show subset subspace Suppose theory topological space u-integrable u-measurable uniformly weak topology weakly compact weakly sequentially compact zero ΕΕΣ